### Abstract

We characterise the stress singularity of the Oldroyd-B, Phan-Thien-Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces.Byintegrating along streamlines,weverify the polymer stress behavior of r^{-4/11} for PTT and r^{-5/16} viscoelastic velocity field. distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity. A three-region asymptotic structure is again apparent with now a polymer stress singularity of r^{-4/5}. The highWeissenberg boundary layer equations are found to manifest themselves at the stick surface and are of thickness r^{3/2}. At the slip surface, dominant balance between the upper convected stress and rate-of-strain terms gives a slip boundary layer of thickness r^{2}. The solution of the slip boundary layer shows that the polymer stress is now singular along the slip surface. These results are supported through numerical integration along streamlines of the Oldroyd-B equations in a Newtonian velocity field. The Oldroyd-B model thus extends the point singularity at the join of the stick and slip surfaces to the whole of slip surface. As such, it does not have a physically meaningful solution in a Newtonian velocity field. We would expect a similar stress behavior for this model in the true viscoelastic velocity field.

Language | English |
---|---|

Article number | 121604 |

Journal | Physics of Fluids |

Volume | 29 |

Issue number | 12 |

Early online date | 16 Oct 2017 |

DOIs | |

Status | Published - 1 Dec 2017 |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*29*(12), [121604]. https://doi.org/10.1063/1.4993782

**Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity.** / Evans, J. D.; Palhares Junior, I. L.; Oishi, C. M.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 29, no. 12, 121604. https://doi.org/10.1063/1.4993782

}

TY - JOUR

T1 - Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity

AU - Evans, J. D.

AU - Palhares Junior, I. L.

AU - Oishi, C. M.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - We characterise the stress singularity of the Oldroyd-B, Phan-Thien-Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces.Byintegrating along streamlines,weverify the polymer stress behavior of r-4/11 for PTT and r-5/16 viscoelastic velocity field. distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity. A three-region asymptotic structure is again apparent with now a polymer stress singularity of r-4/5. The highWeissenberg boundary layer equations are found to manifest themselves at the stick surface and are of thickness r3/2. At the slip surface, dominant balance between the upper convected stress and rate-of-strain terms gives a slip boundary layer of thickness r2. The solution of the slip boundary layer shows that the polymer stress is now singular along the slip surface. These results are supported through numerical integration along streamlines of the Oldroyd-B equations in a Newtonian velocity field. The Oldroyd-B model thus extends the point singularity at the join of the stick and slip surfaces to the whole of slip surface. As such, it does not have a physically meaningful solution in a Newtonian velocity field. We would expect a similar stress behavior for this model in the true viscoelastic velocity field.

AB - We characterise the stress singularity of the Oldroyd-B, Phan-Thien-Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces.Byintegrating along streamlines,weverify the polymer stress behavior of r-4/11 for PTT and r-5/16 viscoelastic velocity field. distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity. A three-region asymptotic structure is again apparent with now a polymer stress singularity of r-4/5. The highWeissenberg boundary layer equations are found to manifest themselves at the stick surface and are of thickness r3/2. At the slip surface, dominant balance between the upper convected stress and rate-of-strain terms gives a slip boundary layer of thickness r2. The solution of the slip boundary layer shows that the polymer stress is now singular along the slip surface. These results are supported through numerical integration along streamlines of the Oldroyd-B equations in a Newtonian velocity field. The Oldroyd-B model thus extends the point singularity at the join of the stick and slip surfaces to the whole of slip surface. As such, it does not have a physically meaningful solution in a Newtonian velocity field. We would expect a similar stress behavior for this model in the true viscoelastic velocity field.

UR - http://www.scopus.com/inward/record.url?scp=85031781882&partnerID=8YFLogxK

U2 - 10.1063/1.4993782

DO - 10.1063/1.4993782

M3 - Article

VL - 29

JO - Physics of Fluids

T2 - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 12

M1 - 121604

ER -